Remark 4.4 Different from those for the linear damped wave equation with mass term on the Heisenberg group studied in [1]}, or the linear damped wave equation on the Heisenberg group studied in [2]}, it is challenging for us compute explicit eigenvalues for the system (REF ) since we are now focusing on a coupled system with a vector unknown rather than a scalar unknown. To overcome this difficulty, in the subsections, we will use another approach to get pairwise distinct eigenvalues and some pointwise estimates for \(W(t,\lambda )_{k,\ell }\)  instead of direct calculating the eigenvalues and their corresponding eigenprojections.

